If 0 ≤ θ  ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is

(a)

\(\frac { 2\pi }{ 3 } \)

(b)

\(\frac { 3\pi }{ 4 } \)

(c)

\(\frac { 5\pi }{ 6 } \)

(d)

\(\frac { \pi }{ 4 } \)

The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is

(a)

\(\cfrac { 1 }{ i+2 } \)

(b)

\(\cfrac { -1 }{ i+2 } \)

(c)

\(\cfrac { -1 }{ i-2 } \)

(d)

\(\cfrac { 1 }{ i-2 } \)

If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \), then (A, B) equals

(a)

(1, 0)

(b)

(−1, 1)

(c)

(0, 1)

(d)

(1, 1)

The least positive integer n such that \(\left( \frac { 2i }{ 1+i } \right) ^{ n }\) is a positive integer is ____________

(a)

16

(b)

8

(c)

4

(d)

2

The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is

(a)

2

(b)

4

(c)

1

(d)

∞

If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________

(a)

0, 2

(b)

0, 1

(c)

0, 3

(d)

0, √3

If cot^-1 2 and cot^-1 3 are two angles of a triangle, then the third angle is

(a)

\(\frac{\pi}{4}\)

(b)

\(\frac{3\pi}{4}\)

(c)

\(\frac{\pi}{6}\)

(d)

\(\frac{\pi}{3}\)

The radius of the circle passing through the point(6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is

(a)

10

(b)

\( {2} \sqrt {5}\)

(c)

6

(d)

4

The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is

(a)

a parabola

(b)

a hyperbola

(c)

an ellipse

(d)

a circle

A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon's angle of elevation in radian per second when the balloon is 30 metres above the ground.

(a)

\(\frac{3}{25} \text { radians } / \mathrm{sec}\)

(b)

\(\frac{4}{25} \text { radians } / \mathrm{sec}\)

(c)

\(\frac{1}{5} \text { radians } / \mathrm{sec}\)

(d)

\(\frac{1}{3} \text { radians } / \mathrm{sec}\)